A Level Mathematics CCEA
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143 topics in 26 modules
☑️ Algebra and Functions (Pure Mathematics) 18 topics
- Indices
- Surds
- Rationalising denominators
- Quadratics: Factorising
- Quadratics: Solving equations
- Quadratics: Completing the square
- Quadratics: Disguised quadratics
- Quadratics: Using discriminant
- Quadratics: Linear and quadratic inequalities
- Simultaneous equations: Linear (with 2 or 3 variables)
- Simultaneous equations: Linear and Quadratic
- Graphs: Quadratics and cubics
- Graphs: Reciprocal functions
- Graphs: Graph Transformations
- Functions: Definition and terminology
- Functions: Composite function
- Functions: Inverse functions and graphs
- Functions: Modulus function
☑️ Partial Fractions (Pure Mathematics) 4 topics
- Simplifying rational expressions by factorising
- Simplifying rational expressions by cancelling
- Simplifying rational expressions by algebraic division
- Decomposing rational functions into partial fractions
☑️ Exponentials and Logarithms (Pure Mathematics) 7 topics
- Properties of the Exponential function
- Properties of Logarithms
- Laws of Logarithms
- Graphs of exponential and log functions
- Relationship between exponential and log functions
- Solution of equations and inequations involving exponential functions
- Exponential growth and decay
☑️ Differentiation (Pure Mathematics) 9 topics
- Interpreting derivative as gradient of curve
- Interpretating derivative as rate of change
- Gradients and stationary points
- Use of 2nd derivative
- Increasing and decreasing functions
- Tangents and normal
- Differentiation of exponential, logarithmic and trig functions
- Chain rule
- Product rule
☑️ Polynomials (Pure Mathematics) 3 topics
- Quotient rule
- Long division
- Use of remainder and factor theorems
☑️ Integration (Pure Mathematics) 7 topics
- Understanding integration as reverse of differentiation
- Definite integrals
- Area between 2 curves
- Integration using Substitution
- Integration using Parts
- Integration using Partial fractions
- Volumes of revolution
☑️ Parametric Equations (Pure Mathematics) 2 topics
- Using parametric equations of curves
- Converting between parametric and Cartesian forms
☑️ Trigonometry (Pure Mathematics) 13 topics
- Sine and cosine rules
- Area of triangle
- Graph of sine
- Graph of cosine
- Graph of tangent
- Radian measure
- Arc length and sector area
- Definitions of secant, cosecant, cotangent
- Definitions of arcsin, arccos and arctan
- Compound angle formulae for sine, cosine and tangent
- Double angle formulae
- Harmonic form
- Constructing proofs involving trig functions and identities
☑️ Circle Geometry (Pure Mathematics) 4 topics
- Equation of a circle
- Finding the centre and radius
- Finding the equation of a tangent at a given point on the circumference
- Using standard circle properties
☑️ Coordinate Geometry (Pure Mathematics) 4 topics
- Equation of a straight line
- Midpoint of a line
- Length of line segment
- Conditions for parallel/perpendicular lines
☑️ Sequences and Series (Pure Mathematics) 5 topics
- Simple sequences, including recurrence relations
- Convergence, divergence and oscillation
- Use of sigma notation for series
- Arithmetic Progressions
- Geometric Progressions
☑️ Vectors (Pure Mathematics) 7 topics
- Using vectors in 2 dimensions
- Calculating magnitude and direction
- Converting between component form and magnitude/direction form
- Adding vectors
- Multiplying by a scalar
- Position vectors
- Distance between 2 points represented by position vectors
☑️ Quantities and Units in Mechanics (Applied Mathematics) 1 topic
- Quantities and Units in the SI system
☑️ Kinematics (Applied Mathematics) 3 topics
- Motion in a straight line
- Motion in two dimensions in vector form
- Constant acceleration
☑️ Forces and Newton's Laws (Applied Mathematics) 9 topics
- Force as a vector
- Resolving forces
- Resultant of forces
- Equilibrium
- Newton's 2nd law
- Using weight
- Using friction (including limiting friction)
- Newton's 3rd law
- Applications of vectors in a plane
☑️ Impulse and Momentum (Applied Mathematics) 6 topics
- Simple use of impulse and momentum
- Conservation of linear momentum
- Direct collisions and explosions
- Associated units and language
- s-t and v-t graphs
- Equations of motion
☑️ Moments (Applied Mathematics) 7 topics
- Rods
- Ladders
- Hinged beams
- Vector format
- Using formula for time of flight
- Using formula for range
- Using equation of path of flight
☑️ Probability (Applied Mathematics) 8 topics
- Using addition and multiplication laws
- Mutually exclusive events
- Exhaustive events
- Statistical dependence/independence
- Conditional probability formula
- Tree diagrams
- Venn diagrams
- 2-way tables
☑️ Sampling (Applied Mathematics) 3 topics
- Basic terminology
- Sampling techniques
- Making inferences with sampling
☑️ Histograms (Applied Mathematics) 1 topic
- Interpreting Histograms
☑️ Statistical Measures (Applied Mathematics) 2 topics
- Mean, median, mode
- Standard deviation and variance
☑️ Correlation (Applied Mathematics) 3 topics
- Scatter graphs and regression lines
- Interpreting correlation
- Product-moment correlation
☑️ Data Presentation and Interpretation (Applied Mathematics) 5 topics
- Single Variable Data
- Bivariate Data
- Measures of average and spread
- Calculations of mean and standard deviation
- Outliers and cleaning data
☑️ Binomial Distribution (Applied Mathematics) 2 topics
- Calculating probabilities using the binomial distribution
- Links to binomial expansion and tree diagrams
☑️ Hypothesis Testing (Applied Mathematics) 7 topics
- The language of hypothesis testing
- Conducting a hypothesis test for the proportion in the binomial distribution
- Conducting a hypothesis test for the mean of a normal distribution
- Interpreting a correlation coefficient using a p-value or critical value
- Using a trapezium rule as an approximation to the area under a curve
- Location of roots
- Newton-Raphson method
☑️ Normal Distribution (Applied Mathematics) 3 topics
- Normal distribution as an example of a continuous probability distribution
- Finding probabilities using the normal distribution
- Binomial/normal models
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A Level Mathematics CCEA Revision Content
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A Level Mathematics CCEA - Algebra and Functions (Pure Mathematics) - Indices Content Preview
Algebra and Functions (Pure Mathematics)
Indices
Basic Rules of Indices
- Any number raised to the power of 1 is the number itself, e.g: a^1 = a
- Any number raised to the power of 0 is 1, e.g: a^0 = 1
- If two terms with the same base are multiplied together, the powers are added, e.g: a^m * a^n = a^(m+n)
- If a term with a base of "a" raised to a power "m" is itself raised to a power "n", multiply the powers, e.g: (a^m)^n = a^(mn)
- If two terms with the same base are divided, subtract the exponent of the denominator from the exponent of the numerator, e.g: a^m / a^n = a^(m-n)
- If a term in the denominator has a negative exponent, it can be moved to the numerator and made positive, e.g: 1 / a^-n = a^n
- The n-th root of a number "a" can be denoted by a ^ (1/n)
Laws of Indices Involving Fractions
- a^-n = 1 / a^n, which means a reciprocal of a number can be written as a negative exponent.
- The n-th root of a number can be expressed as a power with a fractional exponent, e.g: n√a = a^(1/n)
- If the power of a term is a fraction, the denominator of the fraction is the root, and the numerator is the power. For example, a^(m/n) = ( n√a ) ^m
Simplification Using Indices
- In order to simplify expressions with indices, use the laws of indices to combine terms.
- Simplify the expression step by step, until no further simplification is possible.
- When dealing with algebraic expressions, it's important to remember that these rules apply only to terms with the same base.
Indices and Surds
- A Surd is an expression that includes a root (√). Surds are dealt with using the laws of indices.
- The method of 'rationalising the denominator' is used to remove surds from the denominator of a fraction.
- To simplify a surd, look for the largest square number which divides into the number under the surd, and use the rule √ab = √a * √b.
Exponential Equations
- Exponential equations are those where the variable is in the exponent.
- They can be solved using the principle that if a^m = a^n, then m = n.
- In cases where this cannot be applied directly, logarithms may be used to solve the equation.
Question: Simplify the following expression using the laws of indices: `(2^3 * 2^-4) / (2^-2)`.
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